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To find the general term for the sequence \(\{a_n\}\) given the conditions:

\[ a_1 = 1 \]
\[ a_{n+1} = a_n + 3^n \quad (n = 1, 2, 3, \ldots) \]

We can start by writing out the first few terms of the sequence to identify any pattern:

1. \( a_1 = 1 \)
2. \( a_2 = a_1 + 3^1 = 1 + 3 = 4 \)
3. \( a_3 = a_2 + 3^2 = 4 + 9 = 13 \)
4. \( a_4 = a_3 + 3^3 = 13 + 27 = 40 \)

Notice that each term adds the next power of 3 to the previous term. To find a pattern, we can express \( a_n \) in a more general form.

Let's write \( a_n \) in terms of the initial value and the sum of the powers of 3:

\[ a_2 = a_1 + 3^1 \]
\[ a_3 = a_1 + 3^1 + 3^2 \]
\[ a_4 = a_1 + 3^1 + 3^2 + 3^3 \]

In general, we can express \( a_n \) as:

\[ a_n = a_1 + 3^1 + 3^2 + \cdots + 3^{n-1} \]

The sum of the series \( 3^0 + 3^1 + 3^2 + \cdots + 3^{n-1} \) is a geometric series with the first term \( a = 1 \) (since \( 3^0 = 1 \)) and common ratio \( r = 3 \). The sum of the first \( n \) terms of a geometric series is given by:

\[ S_n = a \frac{r^n - 1}{r - 1} \]

For our series, \( a = 1 \) and \( r = 3 \), so:

\[ S_n = \frac{3^n - 1}{3 - 1} = \frac{3^n - 1}{2} \]

Therefore, the general term \( a_n \) can be written as:

\[ a_n = 1 + \frac{3^n - 3}{2} \]

Simplifying further:

\[ a_n = 1 + \frac{3^n - 3}{2} = \frac{2 + 3^n - 3}{2} = \frac{3^n - 1}{2} \]

Thus, the general term for the sequence is:

\[ \boxed{a_n = \frac{3^n - 1}{2}} \]

Would you like more details or have any questions? Here are some questions you might want to ask next:

1. How can I prove the formula for the general term of a sequence?
2. Can you show me the derivation of the sum of a geometric series?
3. What are other common types of sequences in mathematics?
4. How can I apply this formula to find a specific term in the sequence?
5. Can you explain more about the properties of geometric sequences?

**Tip:** Always check the initial conditions and a few terms to verify your derived general formula.

Learn how to find the general term of a sequence with specific initial conditions and a recursive formula. This problem involves sequences and series, focusing on the sum of a geometric series. Suitable for high school students studying sequences in mathematics.

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